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\begin{document}

\pagestyle{fancy}
\fancyhead{}
\lhead{Haojiong Shangguan}
\chead{Math}
\rhead{2021/6/9}

\section*{1.Introduction to multigrid method}
The multigrid method is a fast iterative method for solving the
discrete equations of partial differential problems on a multi-layer
grid. The basic idea is sweeping the error and correcting the initial
guess by convertion between fine and coarse grid, which can improve
the convegence rate effectively. 

\section*{2.The principle of the method}
First give an initial guess and iterate on a fine grid to make the HF errors damp
and the LF errors remain. Then, the LF part of the error
on the fine grid can be converted to HF part of the error on the
coarse grid by restriction operator,
which can be rapidly damped by iterating on the coarse
grid. Finally, in order to ensure the accuracy, the results of the
iteration on the coarse grid are interpolated to the fine grid, which
can correct the initial guess.

\section*{3.Diffrent types of multigrid method}
The complete process is in the notes.\\

(1)V-cycle scheme(VC)\\

Main process:iteration method(Jacobi, Gauss-Seidel, etc), restriction
operator(full-weigthing, injection, etc) and interpolation
operator(linear, quadratic, etc)\\

(2)Full multigrid V-cycle scheme(FMG)\\

Main process:V-cycle, restriction
operator(full-weigthing, injection, etc) and interpolation
operator(linear, quadratic, etc)\\

\section*{4.Application of multigrid method}
We can apply multigrid method to solve one-dimensional Possion equation
\begin{equation*}
  -u''(x) = f(x),
\end{equation*}
on $\Omega = [0,1]$ with the Dirichlet boundary condition $u(0) = u_0$ and
$u(1) = u_1$.

\subsection*{Implementation}
Discretize $\Omega$ by a Cartesian grid with uniform spacing and
approximate the second derivative $u''$ with a centered difference and
we get a linear system
\begin{equation*}
  A\mathbf{u} = \mathbf{f}.
\end{equation*}

The error of a multigrid method is
\begin{equation*}
  \mathbf{e} = \mathbf{u} - \mathbf{\tilde{u}},
\end{equation*}
and the residual of an approximate solution is
\begin{equation*}
  \mathbf{r} = \mathbf{f} - A\mathbf{u}.
\end{equation*}

From the definition of error and residual we can get the error and the
residual of an approximate solution satisfy the residual equation
\begin{equation*}
  A\mathbf{e} = \mathbf{r}.
\end{equation*}

Therefore we can solve the residual equation instead of the linear
system. In other words, these two equations are equivalent.\\

Write a C++ package to implement the VC and FMG solver and then solve
the residual equation.

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